Dirac-Breit theory

The explicit form of the property operators is derived in two steps. First, the scalar and vector fields are determined and, second, they are inserted into the Dirac-Breit [951,952] or any other quasi-relativistic operator derived from it such as the external-field-containing DKH operator, the external-field-containing ZORA Hamiltonian or the similar Breit-Pauli Hamiltonian. The theory of NMR parameters has been derived by Ramsey [953-955] from the nonrelativistic perspective and by Pyykko from the relativistic perspective [956,957]. 

Table 8 Second-order many-body perturbation theory corrections to beryllium-like ions using non-relativistic (E ), Dirac-Coulomb (E ) and Dirac-Coulomb-Breit (E ) hamiltonians, obtained using the atomic precursor to BERTHA, known as SWIRLES. Basis sets are even-tempered S-spinors of dimension N= 17, with exponent sets, Xi generated by Xi = abi-i, with a = 0.413, and p = 1.376. Angular momenta in the range 0 < / < 6 have been included in the partial wave expansion of each second-order energy, and the total relativistic correction toE has been collected as Ef. All energies in hartree.

A full account of the theory of relativistic molecular structure based on standard QED in the Furry picture will be found in a number of publications such as [7, Chapter 22], [8, Chapter 3]. These accounts use a relativistic second quantized formalism. For present purposes, it is sufficient to present the structure of BERTHA in terms of the unquantized effective Dirac-Coulomb-Breit (DCB) A-electron Hamiltonian ... 

The Breit-Pauli Hamiltonian is an approximation up to 1/c2 to the Dirac-Coulomb-Breit Hamiltonian obtained from a free-particle Foldy-Wouthuysen transformation. Because of the convergence issues mentioned in the preceding section, the Breit-Pauli Hamiltonian may only be employed in perturbation theory and not in a variational procedure. The derivation of the Breit-Pauli Hamiltonian is tedious (21). 

The most unsatisfactory features of our derivation of the molecular Hamiltonian from the Dirac equation stem from the fact that the Dirac equation is, of course, a single particle equation. Hence all of the inter-electron terms have been introduced by including the effects of other electrons in the magnetic vector and electric scalar potentials. A particularly objectionable aspect is the inclusion of electron spin terms in the magnetic vector potential A, with the use of classical field theory to derive the results. It is therefore of interest to examine an alternative development and in this section we introduce the Breit Hamiltonian [16] as the starting point. We eventually arrive at the same molecular Hamiltonian as before, but the derivation is more satisfactory, although fundamental difficulties are still present. 

In chapter 3 we showed how the relativistic Breit Hamiltonian can be reduced to non-relativistic form by means of a Foldy Wouthysen transformation. We obtained equations (3.244) and (3.245) which represent the non-relativistic Hamiltonian for two particles of masses m, and nij and electrostatic charges and —ej and from this Hamiltonian we were able to derive the interelectronic interactions. We could, however, consider using (3.244) and (3.245) as the Hamiltonian for an electron of charge —e, = — e and mass m, = m, and a nucleus of mass nij = Ma and charge —ej = + 7. e. As before, we make the assumption that the nucleus has spin 1 /2, behaves like a Dirac particle and has an anomalous magnetic moment compared with that given by the Dirac theory. Consequently we may rewrite (3.245) by making the replacements... 

Other electron nuclear interaction terms involving 7ra rather than Ia arise from this treatment. However, these terms have all been dealt with in the previous chapter and we do not repeat them here.) The terms in (4.23) are the same as those obtained previously starting from the Dirac equation. Equation (3.244) will yield both the electron and nuclear Zeeman terms and a Breit equation for two nuclei, reduced to non-relativistic form, would yield the nuclear-nuclear interaction terms. Although many nuclei have spins other than 1/2, and even the proton with spin 1 /2 has an anomalous magnetic moment which does not fit the simple Dirac theory, the approach outlined here is fully endorsed by quantum electrodynamics provided that only terms involving M l are retained (see equation (4.23)). The interested reader is referred to Bethe and Salpeter [11] for further details. In our present application we see that the expressions for both... 

The fundamental expressions which describe the interaction of an external magnetic field with the electrons and nuclei within a molecule were developed from the Dirac and Breit equations in chapters 3 and 4. In this section we develop the theory again, making use of the approach described by Flygare [107]. We start with the classical description of the interaction of a free particle of mass m and charge q with an electromagnetic... 

After a general introduction, the methods used to separate nuclear and electronic motions are described. Brown and Carrington then show how the fundamental Dirac and Breit equations may be developed to provide comprehensive descriptions of the kinetic and potential energy terms which govern the behaviour of the electrons. One chapter is devoted solely to angular momentum theory and another describes the development of the so-called effective Hamiltonian used to analyse and understand the experimental spectra of diatomic molecules. The remainder of the book concentrates on experimental methods. 

SO coupling is a relativistic effect. The theory of the interaction of the magnetic moments of the electron spin and the orbital motion in one- and two-electron atoms has been formulated independently by Heisenberg and Pauli [12,13], shortly before the advent of the four-component Dirac theory of the electron [14]. Breit later has added the retardation correction [15]. The resulting Breit-Pauli SO operator, which can more elegantly be derived from the Dirac equation via a Foldy-Wouthuysen transformation [16], was thus well known for atoms since the early 1930s [17]. 

In Table 6.3, the values of De for RfCU are compared with those obtained within various approximations using relativistic effective core potentials (RECP) Kramers-restricted Hartree-Fock (KRHF) (Han et al 1999), averaged RECP including second-order M0ller-Plesset perturbation theory (AREP-MP2) for the correlation part (Han et al. 1999), RECP coupled-cluster single double (triple) [CCSD(T)] excitations (Han et al. 1999), and a Dirac-Fock-Breit (DFB) method (Malli and Styszynski 1998). The AREP-MP2 calculation of De gives 20.4 eV, while the RECP-CCSD(T) method with correlation leads to 18.8 eV. Our value of De of 19.5 eV is just between these calculated values. 

The fine structure was calculated by Pirenne [110] and independently by Berestetski [4J. Minor errors are corrected, and numerical results are given by Ferrell [45]. The approach used by these authors is to write down the Dirac equations for the two particles, and the interaction terms as they are expressed in quantum field theory. The equations can be transformed so that the particle spins appear explicitly. The interaction terms are found to comprise the Coulomb energy, the Breit interaction, and a term analogous to the Fermi expression for... 

Aspects of the relativistic theory of quantum electrodynamics are first reviewed in the context of the electronic structure theory of atoms and molecules. The finite basis set parametrization of this theory is then discussed, and the formulation of the Dirac-Hartree-Fock-Breit procedure presented with additional detail provided which is specific to the treatment of atoms or molecules. Issues concerned with the implementation of relativistic mean-field methods are outlined, including the computational strategies adopted in the BERTHA code. Extensions of the formalism are presented to include open-shell cases, and the accommodation of some electron correlation effects within the multi-configurational Dirac-Hartree-Fock approximation. We conclude with a survey of representative applications of the relativistic self-consistent field method to be found in the literature. 

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